When Napier first introduced Logarithms back in the 1600s, he thought that this was a neat little way to do calculations and increase mathematical efficiency. It was not until Euler came along and introduced the two most frequently used forms of Logarithms: — 1) Natural and 2) Common.
When we talk about a logarithm to the base ‘e’, we call it the natural logarithm of that number. It is commonly represented as ln(x) for some positive number x¹.
When someone mentions a common Logarithm, it is the logarithm to the base of ‘10’. It is commonly represented as:
The Common Logarithm is neat. It lets us think about all numbers as powers of 10 and hence lets us use 10 as the basis of a number system. Then why did Euler name the logarithm to the base e as the Natural Logarithm, and not the logarithm to the base 10? Here we loot a couple of scenarios, taken from physics as well as mathematics to justify Euler’s decision.
The Physics Makes Sense!
When we structured our number system, we made it so that every important thing was in some form of 10. Think about millions and billions, they are powers of 10 as well. Some say that it was structured because we have 10 fingers and hence the most elementary form of counting can be exploited to the fullest. Maybe that is why it is instinct for us to hold 10 in such a high regard, and we have! We gave it the title of the “Common Logarithm” after all! But when certain phenomena were observed in the study of Physics and nature, ‘e’ was prioritized simply because it showed up everywhere. Let’s take a look at a few scenarios.
Consider that you are a nuclear scientist who is working on a fission reactor. Given that, you know how many nuclei are there in the fission tank at a moment, can you calculate how many will be left after some time ‘t’ has elapsed?
Well, it was observed that the rate of decay of the Nuclei depends on the number of Nuclei present in the tank. Or,
where N is the number of Nuclei at a given instant of time.
To remove the proportionality sign, we multiply a constant, say lambda, and we multiply with a negative to show that it is decaying.
Now doing some (not so) fancy rearrangement, we get:
Integrating both sides², Putting the bounds on t from 0 to any time t, we get:
Solving using the definition of logarithm,
Where N0 is the number of Nuclei at the start of the fission or the number of Nuclei that are known. Clearly, this is an exponential Law in ‘e’.
This law is called the Nuclear Decay law.
Capacitors and Batteries
Consider you are an electrical engineer. Suppose you are shown the following circuit:
This is called an RC circuit since it features both a Resistance(R) and a capacitor(C). You are to calculate the charge on the plates of the capacitor at any given time ‘t’.
Applying Kirchoff’s Voltage Law³ on the entire Circuit⁴,
where V is the voltage supplied by the battery, i the current and q is the charge at any instant. Now, the current flowing is the same in both the resistance and the capacitor, and since the current is the rate of flow of charge, we can modify our equation to become:
Keeping the like terms on one side, we get:
Now, the maximum charge the capacitor can hold is CV. Let us call that q0 or q nought.
Now integrating, putting the bounds on t from 0 to any time t:
further solving, using the definition of logarithm, we get:
Once again, an exponential relation in terms of ‘e’.
These circuits are used extensively in chargers and electric appliances to rectify the power supply of alternating current to direct current.
Consider that you are a chemist studying the following reaction:
This is an example of a first-order reaction, where the rate of reaction is directly proportional to the concentration of the reactant, for simplicity’s sake, say R.
You are to find out the concentration of the reactant at any time ‘t’ during the course of the reaction.
We know that:
where ROR is the rate of reaction. But since the rate of Reaction is just the change in concentration of reactant over time,
again, introducing a constant and rearranging,
Integrating from t=0 to t=t,
which, as it turns out is the same exact relation as the one we obtained in Nuclear Decay. Hence all nuclear Decays are first-order reactions as well. In fact, almost all the reactions we see in nature are first-order reactions! Here, we also find the relation is exponential in terms of ‘e’. (Conversion of the logarithmic to the exponential form is left as an exercise to the reader).
The conclusion from a Physics Point of View
From the three examples stated above, it must be clear that the number e is involved in many natural processes. Many more examples can be brought up: the Rate Law of Population⁵, The Damped Oscillations Law⁶ and the Law of Atmospheres⁷. Including all of these would make the content repetitive, confusing and a lot more daunting. So we’ll skip them for now. (If you’re interested, I’ll even link some reading at the end!). The conclusion drawn is that the number e is behind many of nature’s processes and observations and hence the logarithm of the base e is called the Natural Logarithm
The Mathematical History
Before Napier, there were mentions of something resembling a logarithm in the works of Gregoire de Saint Vincent. In his famous work in which he quadrized⁸ a rectangular hyperbola, he mentions several properties of the hyperbola which are similar to that of a logarithm.
Christiaan Huygens (he is the one who proposed the wave model of light), and James Gregory later hypothesized a new function called ln(x). Sometime later, Leibniz also managed to integrate dx/x and found a striking resemblance to ln(x).
However, the number ‘e’ and its logarithm were named by none other than Leonard Euler. When working on the same problem as Saint Vincent, he found that the point 2.781…. and its reciprocal lay on the hyperbola xy=1, the same hyperbola used by Saint Vincent, and the area below the hyperbola from (0,0) up to that point, bounded by the vertical asymptote or the y axis was 1 square unit. He thus named the number 2.781…. as ‘e’ (people think that Euler was some kind of narcissist who named a constant in his own name, but it turns out he was biased towards vowels, and ‘a’ was a variable he had already used at the time so he arbitrally called it ‘e’, the next vowel) and christened the logarithm to the base of ‘e’ as the natural logarithm.
Not long after this, Euler arrived at Euler’s Identity, from which the world’s most beautiful equation is obtained.
Interestingly, Roger Coates, the proofreader of Newton’s Principia, and the discoverer of the Newton-Coates Quadrature Formula, arrived at a similar conclusion:
Notes, Conclusions and References
1: Since exponential functions are only defined for positive numbers, logarithmic functions are also defined for positive numbers as well.
2. Integration of dx/x is ln(x).
3. Kirchoff’s Voltage Law states that in a circuit, if you go around in a loop, the potential difference across the same points remains zero, or there is no net voltage around a loop of a circuit.
4. This is a direct consequence of Ohm’s Law (V=iR) and the universal law of capacitors (q=cV)
5. Rate Law of Populations (aka Natural Law of Growth):
6. Law of Damped Oscillations: https://byjus.com/jee/damped-oscillation/
7. Law of Atmospheres: https://en.wikipedia.org/wiki/Barometric_formula
8. Quadrizing a shape means to make the area of a certain shape equal to the area of a square of given side. A very famous example of this is Squaring the Circle, which means to make the area of a circle equal to the area of the square it is inscribed in. It is Quadrizing a circle.
Thank You! If you find a mistake, or have a question, feel free to use the comments! I hope you have a nice day! :p